Calculus Examples

Find the 2nd Derivative f(x)=x square root of 4-x^2
Step 1
Find the first derivative.
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Step 1.1
Use to rewrite as .
Step 1.2
Differentiate using the Product Rule which states that is where and .
Step 1.3
Differentiate using the chain rule, which states that is where and .
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Step 1.3.1
To apply the Chain Rule, set as .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Replace all occurrences of with .
Step 1.4
To write as a fraction with a common denominator, multiply by .
Step 1.5
Combine and .
Step 1.6
Combine the numerators over the common denominator.
Step 1.7
Simplify the numerator.
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Step 1.7.1
Multiply by .
Step 1.7.2
Subtract from .
Step 1.8
Combine fractions.
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Step 1.8.1
Move the negative in front of the fraction.
Step 1.8.2
Combine and .
Step 1.8.3
Move to the denominator using the negative exponent rule .
Step 1.8.4
Combine and .
Step 1.9
By the Sum Rule, the derivative of with respect to is .
Step 1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.11
Add and .
Step 1.12
Since is constant with respect to , the derivative of with respect to is .
Step 1.13
Differentiate using the Power Rule which states that is where .
Step 1.14
Combine fractions.
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Step 1.14.1
Multiply by .
Step 1.14.2
Combine and .
Step 1.14.3
Combine and .
Step 1.15
Raise to the power of .
Step 1.16
Raise to the power of .
Step 1.17
Use the power rule to combine exponents.
Step 1.18
Add and .
Step 1.19
Factor out of .
Step 1.20
Cancel the common factors.
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Step 1.20.1
Factor out of .
Step 1.20.2
Cancel the common factor.
Step 1.20.3
Rewrite the expression.
Step 1.21
Move the negative in front of the fraction.
Step 1.22
Differentiate using the Power Rule which states that is where .
Step 1.23
Multiply by .
Step 1.24
To write as a fraction with a common denominator, multiply by .
Step 1.25
Combine the numerators over the common denominator.
Step 1.26
Multiply by by adding the exponents.
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Step 1.26.1
Use the power rule to combine exponents.
Step 1.26.2
Combine the numerators over the common denominator.
Step 1.26.3
Add and .
Step 1.26.4
Divide by .
Step 1.27
Simplify .
Step 1.28
Subtract from .
Step 1.29
Reorder terms.
Step 2
Find the second derivative.
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Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Multiply the exponents in .
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Step 2.2.1
Apply the power rule and multiply exponents, .
Step 2.2.2
Cancel the common factor of .
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Step 2.2.2.1
Cancel the common factor.
Step 2.2.2.2
Rewrite the expression.
Step 2.3
Simplify.
Step 2.4
Differentiate.
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Step 2.4.1
By the Sum Rule, the derivative of with respect to is .
Step 2.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Multiply by .
Step 2.4.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.6
Add and .
Step 2.5
Differentiate using the chain rule, which states that is where and .
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Step 2.5.1
To apply the Chain Rule, set as .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Replace all occurrences of with .
Step 2.6
To write as a fraction with a common denominator, multiply by .
Step 2.7
Combine and .
Step 2.8
Combine the numerators over the common denominator.
Step 2.9
Simplify the numerator.
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Step 2.9.1
Multiply by .
Step 2.9.2
Subtract from .
Step 2.10
Combine fractions.
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Step 2.10.1
Move the negative in front of the fraction.
Step 2.10.2
Combine and .
Step 2.10.3
Move to the denominator using the negative exponent rule .
Step 2.11
By the Sum Rule, the derivative of with respect to is .
Step 2.12
Since is constant with respect to , the derivative of with respect to is .
Step 2.13
Differentiate using the Power Rule which states that is where .
Step 2.14
Multiply by .
Step 2.15
Since is constant with respect to , the derivative of with respect to is .
Step 2.16
Simplify terms.
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Step 2.16.1
Add and .
Step 2.16.2
Combine and .
Step 2.16.3
Combine and .
Step 2.16.4
Factor out of .
Step 2.17
Cancel the common factors.
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Step 2.17.1
Factor out of .
Step 2.17.2
Cancel the common factor.
Step 2.17.3
Rewrite the expression.
Step 2.18
Move the negative in front of the fraction.
Step 2.19
Multiply by .
Step 2.20
Multiply by .
Step 2.21
Simplify.
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Step 2.21.1
Simplify the numerator.
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Step 2.21.1.1
Rewrite using the commutative property of multiplication.
Step 2.21.1.2
Multiply by .
Step 2.21.1.3
Factor out of .
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Step 2.21.1.3.1
Factor out of .
Step 2.21.1.3.2
Factor out of .
Step 2.21.1.3.3
Factor out of .
Step 2.21.1.4
To write as a fraction with a common denominator, multiply by .
Step 2.21.1.5
Combine and .
Step 2.21.1.6
Combine the numerators over the common denominator.
Step 2.21.1.7
Rewrite in a factored form.
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Step 2.21.1.7.1
Factor out of .
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Step 2.21.1.7.1.1
Factor out of .
Step 2.21.1.7.1.2
Factor out of .
Step 2.21.1.7.1.3
Factor out of .
Step 2.21.1.7.2
Combine exponents.
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Step 2.21.1.7.2.1
Multiply by by adding the exponents.
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Step 2.21.1.7.2.1.1
Move .
Step 2.21.1.7.2.1.2
Use the power rule to combine exponents.
Step 2.21.1.7.2.1.3
Combine the numerators over the common denominator.
Step 2.21.1.7.2.1.4
Add and .
Step 2.21.1.7.2.1.5
Divide by .
Step 2.21.1.7.2.2
Simplify .
Step 2.21.1.8
Simplify the numerator.
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Step 2.21.1.8.1
Apply the distributive property.
Step 2.21.1.8.2
Multiply by .
Step 2.21.1.8.3
Multiply by .
Step 2.21.1.8.4
Subtract from .
Step 2.21.1.8.5
Add and .
Step 2.21.2
Combine terms.
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Step 2.21.2.1
Rewrite as a product.
Step 2.21.2.2
Multiply by .
Step 2.21.2.3
Multiply by by adding the exponents.
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Step 2.21.2.3.1
Multiply by .
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Step 2.21.2.3.1.1
Raise to the power of .
Step 2.21.2.3.1.2
Use the power rule to combine exponents.
Step 2.21.2.3.2
Write as a fraction with a common denominator.
Step 2.21.2.3.3
Combine the numerators over the common denominator.
Step 2.21.2.3.4
Add and .
Step 3
Find the third derivative.
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Multiply the exponents in .
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Step 3.3.1
Apply the power rule and multiply exponents, .
Step 3.3.2
Cancel the common factor of .
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Step 3.3.2.1
Cancel the common factor.
Step 3.3.2.2
Rewrite the expression.
Step 3.4
Differentiate using the Product Rule which states that is where and .
Step 3.5
Differentiate.
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Step 3.5.1
By the Sum Rule, the derivative of with respect to is .
Step 3.5.2
Differentiate using the Power Rule which states that is where .
Step 3.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.4
Add and .
Step 3.6
Raise to the power of .
Step 3.7
Raise to the power of .
Step 3.8
Use the power rule to combine exponents.
Step 3.9
Differentiate using the Power Rule.
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Step 3.9.1
Add and .
Step 3.9.2
Differentiate using the Power Rule which states that is where .
Step 3.9.3
Simplify by adding terms.
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Step 3.9.3.1
Multiply by .
Step 3.9.3.2
Add and .
Step 3.10
Differentiate using the chain rule, which states that is where and .
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Step 3.10.1
To apply the Chain Rule, set as .
Step 3.10.2
Differentiate using the Power Rule which states that is where .
Step 3.10.3
Replace all occurrences of with .
Step 3.11
To write as a fraction with a common denominator, multiply by .
Step 3.12
Combine and .
Step 3.13
Combine the numerators over the common denominator.
Step 3.14
Simplify the numerator.
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Step 3.14.1
Multiply by .
Step 3.14.2
Subtract from .
Step 3.15
Combine fractions.
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Step 3.15.1
Combine and .
Step 3.15.2
Combine and .
Step 3.16
By the Sum Rule, the derivative of with respect to is .
Step 3.17
Since is constant with respect to , the derivative of with respect to is .
Step 3.18
Differentiate using the Power Rule which states that is where .
Step 3.19
Multiply by .
Step 3.20
Since is constant with respect to , the derivative of with respect to is .
Step 3.21
Combine fractions.
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Step 3.21.1
Add and .
Step 3.21.2
Multiply by .
Step 3.21.3
Combine and .
Step 3.21.4
Multiply by .
Step 3.21.5
Combine and .
Step 3.22
Raise to the power of .
Step 3.23
Raise to the power of .
Step 3.24
Use the power rule to combine exponents.
Step 3.25
Add and .
Step 3.26
Factor out of .
Step 3.27
Cancel the common factors.
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Step 3.27.1
Factor out of .
Step 3.27.2
Cancel the common factor.
Step 3.27.3
Rewrite the expression.
Step 3.27.4
Divide by .
Step 3.28
Factor out of .
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Step 3.28.1
Reorder and .
Step 3.28.2
Factor out of .
Step 3.28.3
Factor out of .
Step 3.28.4
Factor out of .
Step 3.29
Cancel the common factor of .
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Step 3.29.1
Cancel the common factor.
Step 3.29.2
Rewrite the expression.
Step 3.30
Simplify.
Step 3.31
Move to the denominator using the negative exponent rule .
Step 3.32
Multiply by by adding the exponents.
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Step 3.32.1
Use the power rule to combine exponents.
Step 3.32.2
To write as a fraction with a common denominator, multiply by .
Step 3.32.3
Combine and .
Step 3.32.4
Combine the numerators over the common denominator.
Step 3.32.5
Simplify the numerator.
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Step 3.32.5.1
Multiply by .
Step 3.32.5.2
Subtract from .
Step 3.33
Combine and .
Step 3.34
Simplify.
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Step 3.34.1
Apply the distributive property.
Step 3.34.2
Apply the distributive property.
Step 3.34.3
Simplify the numerator.
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Step 3.34.3.1
Simplify each term.
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Step 3.34.3.1.1
Expand using the FOIL Method.
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Step 3.34.3.1.1.1
Apply the distributive property.
Step 3.34.3.1.1.2
Apply the distributive property.
Step 3.34.3.1.1.3
Apply the distributive property.
Step 3.34.3.1.2
Simplify and combine like terms.
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Step 3.34.3.1.2.1
Simplify each term.
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Step 3.34.3.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.34.3.1.2.1.2
Multiply by by adding the exponents.
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Step 3.34.3.1.2.1.2.1
Move .
Step 3.34.3.1.2.1.2.2
Use the power rule to combine exponents.
Step 3.34.3.1.2.1.2.3
Add and .
Step 3.34.3.1.2.1.3
Multiply by .
Step 3.34.3.1.2.1.4
Multiply by .
Step 3.34.3.1.2.1.5
Multiply by .
Step 3.34.3.1.2.1.6
Multiply by .
Step 3.34.3.1.2.2
Add and .
Step 3.34.3.1.3
Apply the distributive property.
Step 3.34.3.1.4
Simplify.
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Step 3.34.3.1.4.1
Multiply by .
Step 3.34.3.1.4.2
Multiply by .
Step 3.34.3.1.4.3
Multiply by .
Step 3.34.3.1.5
Multiply by by adding the exponents.
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Step 3.34.3.1.5.1
Move .
Step 3.34.3.1.5.2
Use the power rule to combine exponents.
Step 3.34.3.1.5.3
Add and .
Step 3.34.3.1.6
Multiply by .
Step 3.34.3.1.7
Multiply by .
Step 3.34.3.1.8
Multiply by .
Step 3.34.3.2
Combine the opposite terms in .
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Step 3.34.3.2.1
Add and .
Step 3.34.3.2.2
Add and .
Step 3.34.3.2.3
Subtract from .
Step 3.34.3.2.4
Subtract from .
Step 3.34.4
Move the negative in front of the fraction.
Step 4
Find the fourth derivative.
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Step 4.1
Differentiate using the Constant Multiple Rule.
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Step 4.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2
Apply basic rules of exponents.
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Step 4.1.2.1
Rewrite as .
Step 4.1.2.2
Multiply the exponents in .
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Step 4.1.2.2.1
Apply the power rule and multiply exponents, .
Step 4.1.2.2.2
Multiply .
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Step 4.1.2.2.2.1
Combine and .
Step 4.1.2.2.2.2
Multiply by .
Step 4.1.2.2.3
Move the negative in front of the fraction.
Step 4.2
Differentiate using the chain rule, which states that is where and .
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Step 4.2.1
To apply the Chain Rule, set as .
Step 4.2.2
Differentiate using the Power Rule which states that is where .
Step 4.2.3
Replace all occurrences of with .
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
Combine and .
Step 4.5
Combine the numerators over the common denominator.
Step 4.6
Simplify the numerator.
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Step 4.6.1
Multiply by .
Step 4.6.2
Subtract from .
Step 4.7
Move the negative in front of the fraction.
Step 4.8
Combine and .
Step 4.9
Simplify the expression.
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Step 4.9.1
Move to the left of .
Step 4.9.2
Move to the denominator using the negative exponent rule .
Step 4.9.3
Multiply by .
Step 4.10
Combine and .
Step 4.11
Multiply by .
Step 4.12
Factor out of .
Step 4.13
Cancel the common factors.
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Step 4.13.1
Factor out of .
Step 4.13.2
Cancel the common factor.
Step 4.13.3
Rewrite the expression.
Step 4.14
By the Sum Rule, the derivative of with respect to is .
Step 4.15
Since is constant with respect to , the derivative of with respect to is .
Step 4.16
Differentiate using the Power Rule which states that is where .
Step 4.17
Multiply by .
Step 4.18
Since is constant with respect to , the derivative of with respect to is .
Step 4.19
Combine fractions.
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Step 4.19.1
Add and .
Step 4.19.2
Combine and .
Step 4.19.3
Multiply by .
Step 4.19.4
Combine and .
Step 4.19.5
Move the negative in front of the fraction.
Step 5
The fourth derivative of with respect to is .